Abstract

An identifying code in a graph G is a subset of vertices with the property that for each vertex \(v \in V(G)\), the collection of elements of C at distance at most 1 from v is non-empty and distinct from the collection of any other vertex. We consider the minimum density \(d^*(\mathcal{S}_k)\) of an identifying code in the square grid \(\mathcal{S}_k\) of height k (i.e. with vertex set \( \mathbb {Z} \times \{1, \dots , k\}\)). Using the Discharging Method, we prove \(\displaystyle \frac{7}{20} + \frac{1}{20k} \le d^*(\mathcal{S}_k) \le \min \left\{ \frac{2}{5}, \frac{7}{20} + \frac{3}{10k} \right\} \), and \(\displaystyle d^*(\mathcal{S}_3) =\frac{7}{18}\).